Spectral reconstruction and representations of finitely generated groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917612509724672 |
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| author | Stessin, Michael |
| author_facet | Stessin, Michael |
| contents | It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general, this is not true for representations of infinite groups, even if they are finitely generated. The goal of this paper is to establish a characterization of representations of finitely generated groups in terms of projective joint spectra. This approach has a clear advantage compared to character classification as it is valid for a much wider family of groups and for both finite and infinite dimensional representations. The main tool in establishing our spectral characterization is a reconstruction of an operator acting on a separable Hilbert space from the proper projective joint spectrum of the quadruple containing this operator along with a certain triple of bounded operators acting on the same space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06836 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral reconstruction and representations of finitely generated groups Stessin, Michael Group Theory Functional Analysis 47A25 It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general, this is not true for representations of infinite groups, even if they are finitely generated. The goal of this paper is to establish a characterization of representations of finitely generated groups in terms of projective joint spectra. This approach has a clear advantage compared to character classification as it is valid for a much wider family of groups and for both finite and infinite dimensional representations. The main tool in establishing our spectral characterization is a reconstruction of an operator acting on a separable Hilbert space from the proper projective joint spectrum of the quadruple containing this operator along with a certain triple of bounded operators acting on the same space. |
| title | Spectral reconstruction and representations of finitely generated groups |
| topic | Group Theory Functional Analysis 47A25 |
| url | https://arxiv.org/abs/2312.06836 |